Fuzzy Translations of A Fuzzy Set in UP-Algebras

Aiyared Iampan

doi:10.22342/jims.23.2.371.1-19

Abstract

In this paper, we apply the concept of fuzzy translations of a fuzzy set to UP-algebras.For any fuzzy set $\mu$ in a UP-algebra,the concepts of fuzzy $\alpha$-translations of $\mu$ of type \textrm{I} and of fuzzy $\beta$-translations of $\mu$ of type \textrm{II} are introduced, their basic properties are investigated and some useful examples are discussed.The concepts of prime fuzzy sets and of weakly prime fuzzy sets in UP-algebras are also studied.Moreover, we discuss the concepts of extensions and of intensions of a fuzzy set in UP-algebras.

Highlights

Fuzzy sets were extended to fuzzy translations in many logical algebras such as: In 2009, Lee, Jun and Doh [19] investigated relations among fuzzy translations, fuzzy extensions and fuzzy multiplications of fuzzy subalgebras in BCK/BCI-algebras
In 2014, Ansari and Chandramouleeswaran [4] introduced the notions of fuzzy translations, fuzzy extensions and fuzzy multiplications of fuzzy β-ideals of β-algebras
Senapati [5] introduced the notions of fuzzy translations, fuzzy extensions and fuzzy multiplications of fuzzy subalgebras in BGalgebras

Summary

Fuzzy α-Translation of a Fuzzy Set

We study the basic properties of UP-subalgebras (resp. UPfilters, UP-ideals, strongly UP-ideals) and fuzzy UP-subalgebras (resp. fuzzy UPfilters, fuzzy UP-ideals, fuzzy strongly UP-ideals) of a UP-algebra, and study the concept of prime and weakly prime of subsets and of fuzzy sets of a UP-algebra. Weakly prime fuzzy UP-filter, weakly prime fuzzy UP-ideal, weakly prime fuzzy strongly UP-ideal) of A, the fuzzy α-translation μ†α of μ is a weakly prime fuzzy UP-subalgebra If there exists α ∈ [0, †] such that the fuzzy α-translation μ†α of μ is a weakly prime fuzzy UP-subalgebra Fuzzy UP-filter, fuzzy UPideal, fuzzy strongly UP-ideal) of A, the fuzzy α-translation μ†α of μ is a fuzzy UP-subalgebra Fuzzy UP-filter, fuzzy UP-ideal, fuzzy strongly UP-ideal) extension ν of the fuzzy β-translation μ†β of μ, there exists α ∈ [0, †] with α ≥ β such that ν is the fuzzy UP-subalgebra

Fuzzy β-Translation of a Fuzzy Set

Conclusions